Integral equations of electromagnetics present a powerful theoretical foundation for solution of the problems of scattering and radiation of electromagnetic waves on complex penetrable, impenetrable and composite structures. Such problems are fundamental in many practical applications such as antenna design, analysis of microwave circuits, prediction of signal propagation of high-speed interconnects, stealth technology and many others. The key advantages offered by the integral equation solutions compared to the direct solution of the differential equations of electromagnetics as described in J.-M. Jin, 2002 and A. Taflove et al., 2005 include (i) localization of the unknown field quantities to the boundaries of the analyzed structures instead of their volumes as discussed in A. Peterson et al., 1998; (ii) exact evaluation of the wave propagation over long distances as discussed in W. C. Chew et al., 2001; and (iii) possibilities for abstraction of a complex environment surrounding the objects such as stratified media as discussed in W. C. Chew, 1999.
Previously known types of the integral equations in electromagnetics include the surface integral equation formulations for the piece-wise homogeneous penetrable and impenetrable scatterers such as electric field integral equation (EFIE), magnetic field integral equation (MFIE), combined field integral equation (CFIE), Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) and Muller equations; volume integral equations for inhomogeneous penetrable scatterers (W. C. Chew et al., 2001); and single-source integral equations (SSIEs) as discussed in D. Swatek, 1999 and Z. G. Qian et al, 2007. Stable variants of the above equations enabling and improving the above-listed classical formulations at low-frequencies as discussed in Z. G. Qian et al., 2009; Z. G. Qian, 2009; M. Taskinen et al., 2006; W. Wu et al., 1994; J. Zhao et al., 2000; and K. Cools, 2011; in the presence of high disparity of the material properties as discussed in P. Yl-Oijala et al., 2005 and J. Markkanen et al., 2012, under conditions of oversampling as discussed in F. Valdes et al., 2008, and in the presence of stratified media discussed in K. A. Michalski et al., 1990 and A. Kishk et al., 1986 have also been developed. In this work we introduce a new type of a single-source integral equation by combining the ideas of surface integral representation of the electromagnetic field developed in the theory of the single-source integral equations and the traditional volume equivalence principle.
The classical surface integral equations on penetrable objects are derived through application of either the surface equivalence principle or through application of the Green's theorem to the volumes inside and/or outside the scatterer. Both approaches lead to the same expressions of the electromagnetic field inside and outside the scatterer regions as discussed in C. T. Tai, 1994. These surface integral expressions can be viewed as superpositions of the electric and magnetic types of elementary spherical waves emanating from the region boundaries and weighed by the tangential components of the true electromagnetic field, commonly referred to as the equivalent electric and magnetic currents. As such, the electromagnetic field inside a source-free region of the scatterer is expressed as a superposition of the two types of waves produced by the electric and magnetic dipoles situated on the boundary, both types of waves satisfy the same homogeneous curl-curl equation as the field they produce as a result of their superposition. Two questions then arise in the process of traditional surface integral equation derivation:                1) Is it necessary to include both types of the waves into the superposition in order to obtain the correct field representation or taking only one type of wave (waves of only the electric currents or waves of only the magnetic currents) suffices?        2) Must the strengths of the electric and magnetic dipoles situated at the boundary be determined by the true values of the tangential components of the fields themselves on the boundary?        
The classical equivalence principle gives us the affirmative answer to both of these questions as discussed in R. Harrington, 1993. The only exception to these rules within the equivalence principle come from the possibilities of eliminating one type of the sources at the expense of making the waves of the other type more complicated. For example, one could take only the fields of the magnetic point sources in the presence of the PEC enclosure formed by the boundary of the object and eliminate the contribution from the equivalent electric currents. The magnitudes of the magnetic dipoles however still must be determined by the values of the tangential component of the true electric field on the boundary of the object.
It turns out, however, that neither of the above two requirements are necessary and greater flexibility in the integral representations of the field is possible than that prescribed by the classical equivalence principle. The breakthrough understanding of this fact came with the paper of D. Maystre and P. Vincent, 1972 who showed that:                1) The weights of the electric and/or magnetic dipoles on the surface of the scatterer do not have to be determined by the tangential components of the true electric and magnetic fields. They can be some arbitrary surface vector functions, i.e. fictitious currents. The only requirement on those fictitious currents is that the superposition of the waves they radiate must add up to the true field at every point throughout the scatterer.        2) It is possible to represent the true field inside a scatterer as either superposition of only the electric dipole fields in free space or only the magnetic dipole fields in free space. Indeed, the combination of the two with arbitrary weights is possible as well. In other words, for as long as the field is build from the waves satisfying the same homogeneous wave equation throughout the volume as the sought field, such representation is able to reproduce the true field. For the concept to be practical though, the question of how to find the magnitudes of the fictitious current sources producing such waves would still remain to be answered.        
In D. Maystre and P. Vincent, 1972, it was proposed to determine the fictitious currents through substitution of their integral field representations into the classical surface equivalence principle. Such substitution forms a traditional single source integral equation with respect to the unknown fictitious current density and constrains the latter. In this work we take a different approach. We constrain the fictitious currents through the volume equivalence principle instead. Namely, we substitute the superposition of the waves produced by the fictitious current into the volume equivalence principle. Traditionally, the volume EFIE with respect to the polarization current is enforced throughout the entire volume of the scatterer due to the volumetric polarization current being unknown in the volume. In the case of field representation as a superposition of the waves produced by a single fictitious current source, the unknown function is a tangential vector function localized to the surface of the scatterer. As a result, upon the substitution of the superposition field representation into the V-EFIE it is sufficient to enforce the latter only at the boundary of the scatterer and only for the tangential component of the total field. Such constraining of the single fictitious current on the scatterer boundary through enforcement of the volume integral equation at the boundary of the scatterer for the tangential component of the field produces a new type of integral equation which we term as the Surface-Volume-Surface (SVS) EFIE. The name of the new equation stems from the occurrence in it of the field translations from the surface of the scatterer to its volume and subsequently from the volume back to the surface.
Previously, as discussed in Menshov et al., 2013, Okhmatovski et al., 2016, and F L. S. Hosseini et al., 2016, the SVS-EFIE has been derived for the scalar and vector cases of 2D scattering under TM-polarization and TE-polarization, respectively.